Derivative pricing using neural networks

Derivative pricing using neural networks#

Over the past few years, advancements in artificial intelligence and machine learning have naturally led to the adoption of these techniques in many other fields. Neural networks, for example, have become a highly flexible tool capable of solving a wide variety of problems such as image classification, speech recognition, and language translation.

Today, we also see their application in various industries, including finance. In this notebook, we demonstrate how these models can be used—for instance, to price derivatives. Although derivative pricing is a relatively simple and well-studied problem, it provides an interesting example to illustrate how neural networks can solve financial problems, not only in pricing but also in inverse problems (e.g., model calibration) and risk management.

Solving differential equations with neural networks#

In this entry, we focus on a particular technique: Physics-Informed Neural Networks (PINNs). As the name suggests, these neural networks are designed to incorporate additional information derived from physical laws or known mathematical relationships. Typically, this is achieved by embedding these laws into the loss function to guide the network toward a solution that satisfies the given differential equation.

What are Physics-Informed Neural Networks (PINNs)?#

First we begin with some definitions. Suppose we wish to solve a partial differential equation (PDE) of the form $ \mathcal{N}[V(\mathbf{x})] = 0,\quad \mathbf{x} \in \Omega, $ where:

$\mathcal{N}$ is a differential operator,
$V(\mathbf{x})$ is the unknown solution,
$\Omega$ is the spatial (and possibly temporal) domain.

In a PINN, we approximate $V(\mathbf{x})$ with a neural network $V_\theta(\mathbf{x})$ parameterized by $\theta$. The loss function is constructed to penalize deviations from both the governing PDE in the domain and the prescribed boundary conditions on $\partial \Omega$. Formally, the loss can be written as:

\[ \mathcal{L}(\theta) = \mathcal{L}_{\mathrm{PDE}}(\theta) + \mathcal{L}_{\mathrm{BC}}(\theta), \]

with

\[ \mathcal{L}_{\mathrm{PDE}}(\theta) = \frac{1}{N_r} \sum_{i=1}^{N_r} \left| \mathcal{N}[V_\theta(\mathbf{x}_r^i)] \right|^2, \]

and

\[ \mathcal{L}_{\mathrm{BC}}(\theta) = \frac{1}{N_b} \sum_{j=1}^{N_b} \left| V_\theta(\mathbf{x}_b^j) - g(\mathbf{x}_b^j) \right|^2, \]

where:

$\{\mathbf{x}_r^i\}_{i=1}^{N_r}$ are the collocation points in the interior of the domain $\Omega$,
$\{\mathbf{x}_b^j\}_{j=1}^{N_b}$ are the points on the boundary $\partial \Omega$ with known values $g(\mathbf{x}_b^j)$.

In order to train the network, we minimize the loss function using gradient-based optimization techniques. The resulting neural network will approximate the solution to the PDE in the domain $\Omega$ and satisfy the boundary conditions on $\partial \Omega$.

The idea is not to overcomplicate things, so we start with the example. First, we import the necessary libraries:

A module that was compiled using NumPy 1.x cannot be run in
NumPy 2.2.3 as it may crash. To support both 1.x and 2.x
versions of NumPy, modules must be compiled with NumPy 2.0.
Some module may need to rebuild instead e.g. with 'pybind11>=2.12'.

If you are a user of the module, the easiest solution will be to
downgrade to 'numpy<2' or try to upgrade the affected module.
We expect that some modules will need time to support NumPy 2.

Traceback (most recent call last):  File "<frozen runpy>", line 198, in _run_module_as_main
  File "<frozen runpy>", line 88, in _run_code
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel_launcher.py", line 18, in <module>
    app.launch_new_instance()
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/traitlets/config/application.py", line 1075, in launch_instance
    app.start()
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelapp.py", line 739, in start
    self.io_loop.start()
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/tornado/platform/asyncio.py", line 205, in start
    self.asyncio_loop.run_forever()
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/asyncio/base_events.py", line 608, in run_forever
    self._run_once()
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/asyncio/base_events.py", line 1936, in _run_once
    handle._run()
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/asyncio/events.py", line 84, in _run
    self._context.run(self._callback, *self._args)
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelbase.py", line 545, in dispatch_queue
    await self.process_one()
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelbase.py", line 534, in process_one
    await dispatch(*args)
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelbase.py", line 437, in dispatch_shell
    await result
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/ipkernel.py", line 362, in execute_request
    await super().execute_request(stream, ident, parent)
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelbase.py", line 778, in execute_request
    reply_content = await reply_content
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/ipkernel.py", line 449, in do_execute
    res = shell.run_cell(
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/zmqshell.py", line 549, in run_cell
    return super().run_cell(*args, **kwargs)
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3077, in run_cell
    result = self._run_cell(
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3132, in _run_cell
    result = runner(coro)
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/async_helpers.py", line 128, in _pseudo_sync_runner
    coro.send(None)
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3336, in run_cell_async
    has_raised = await self.run_ast_nodes(code_ast.body, cell_name,
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3519, in run_ast_nodes
    if await self.run_code(code, result, async_=asy):
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3579, in run_code
    exec(code_obj, self.user_global_ns, self.user_ns)
  File "/var/folders/cp/l432p0ns1t38qmnyr_3l3r000000gn/T/ipykernel_62331/4220564520.py", line 1, in <module>
    import torch
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/__init__.py", line 1477, in <module>
    from .functional import *  # noqa: F403
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/functional.py", line 9, in <module>
    import torch.nn.functional as F
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/nn/__init__.py", line 1, in <module>
    from .modules import *  # noqa: F403
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/nn/modules/__init__.py", line 35, in <module>
    from .transformer import TransformerEncoder, TransformerDecoder, \
  File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/nn/modules/transformer.py", line 20, in <module>
    device: torch.device = torch.device(torch._C._get_default_device()),  # torch.device('cpu'),
/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/nn/modules/transformer.py:20: UserWarning: Failed to initialize NumPy: _ARRAY_API not found (Triggered internally at /Users/runner/work/pytorch/pytorch/pytorch/torch/csrc/utils/tensor_numpy.cpp:84.)
  device: torch.device = torch.device(torch._C._get_default_device()),  # torch.device('cpu'),

<torch._C.Generator at 0x115567790>

Important

The choice of collocation points is critical for the convergence of the method. One potential improvement is to use quasi-random numbers to distribute the collocation points more uniformly across the domain, thereby avoiding clustering that can occur with some random number generators.

Loss functions and the governing equation#

For the problem of pricing options, the fundamental PDE we need to solve is the Black-Scholes equation. This equation is a parabolic PDE that describes the evolution of the price $V(S,t)$ of an option in terms of time $t$ and the underlying asset price $S$. It is given by:

\[ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0, \]

where:

$V(S,t)$ is the option price,
$S$ is the price of the underlying asset,
$r$ is the risk-free interest rate,
$\sigma$ is the volatility of the underlying asset.

Our objective is to use this PDE to guide the training of our neural network. Modern machine learning libraries, such as PyTorch, offer automatic differentiation capabilities. For example, torch.autograd.grad efficiently computes the necessary derivatives, such as $\nabla V(S,t)$, at each training point. In our fomulation, $\mathcal{L}_{\mathrm{PDE}}(\theta)$ would be represented by the following function:

def pde_loss_f(model, inputs, sigma, r):
    inputs.requires_grad_(True)
    V = model(inputs)

    # First order
    gradients = grad(V, inputs, grad_outputs=torch.ones_like(
        V), create_graph=True)[0]
    dVdt = gradients[:, 0]
    dVdS = gradients[:, 1]

    # Second order
    d2VdS2 = grad(dVdS, inputs, grad_outputs=torch.ones_like(
        dVdS), create_graph=True)[0][:, 1]
    S = inputs[:, 1]
    V = V[:, 0]

    # PDE
    pde_residual = dVdt + 0.5 * sigma ** 2 * S ** 2 * d2VdS2 + r * S * dVdS - r * V
    loss_pde = torch.mean(pde_residual ** 2)
    return loss_pde

Additionally, to incorporate the boundary conditions, we need to define an additional loss functions that force the network to satisfy these constraints. Since the boundary conditions we are using are of Dirichlet type, the loss function is relatively simple (it merely compares the network outputs against the ground truth). However, nothing prevents us from employing other types of boundary conditions (such as Neumann or Robin).

def boundary_loss_f(model, inputs, outputs):
    V_pred = model(inputs)
    boundary_loss = torch.mean((V_pred - outputs) ** 2)
    return boundary_loss

Collocation points#

To solve the PDE using PINNs, we need to define the numerical domain over which the equation will be solved. Much like traditional numerical methods (e.g., finite differences), we select a set of points -called collocation points- where the PDE is enforced. These include points in the interior of the domain $\Omega$ as well as on its boundary $\partial\Omega$.

To enforce the boundary conditions, additional loss terms are added that force the network to satisfy them. For a European call option, the typical boundary conditions are:

At $S = 0$: $$ V(0, t) = 0, $$ since if the underlying asset’s price is zero, the option is worthless.
For $S \to \infty$: $$ V(S, t) \approx S - K e^{-r(T-t)}, $$ as the option behaves like a linear payoff for large $S$.
At expiration $t = T$: $$ V(S, T) = \max(S - K,\, 0), $$ which is simply the payoff of the option.

We sample this points using random numbers, and we will use them to train the neural network. Next is an implementation for sampling this points:

def payoff(s, k):
    return np.maximum(s-k, 0)


def interior_samples(option_config, scaling=4):
    t = np.random.uniform(
        0, option_config['maturity'], option_config['n_samples'])
    s = np.random.uniform(
        0, scaling*option_config['strike'], option_config['n_samples'])
    tag = np.zeros(option_config['n_samples'])
    output = np.zeros(option_config['n_samples'])
    return t, s, tag, output


def top_boundary_samples(option_config, scaling=4):
    t = np.random.uniform(
        0, option_config['maturity'], option_config['n_samples'])
    s = np.ones(option_config['n_samples']) * scaling*option_config['strike']
    strike = option_config['strike']
    s_max = scaling*option_config['strike']
    r = option_config['r']
    T = option_config['maturity']
    output = s_max - strike * np.exp(-r*(T-t))
    tag = np.ones(option_config['n_samples'])
    return t, s, tag, output


def bottom_boundary_samples(option_config):
    t = np.random.uniform(
        0, option_config['maturity'], option_config['n_samples'])
    s = np.zeros(option_config['n_samples'])
    output = np.zeros(option_config['n_samples'])
    tag = np.ones(option_config['n_samples'])
    return t, s, tag, output


def initial_condition_samples(option_config, scaling=4):
    t = np.ones(option_config['n_samples']) * option_config['maturity']
    s = np.random.uniform(
        0, scaling*option_config['strike'], option_config['n_samples'])
    tag = np.ones(option_config['n_samples'])
    output = payoff(s, option_config['strike'])
    return t, s, tag, output

Of course, we need to define the financial parameters that will be used in the problem. For this example, we will use the following values:

config = {'strike': 15, 'maturity': 1,
          'r': 0.04, 'sigma': 0.25, 'n_samples': 1000}

# Generate samples
inner_samples = interior_samples(config)
top_samples = top_boundary_samples(config)
bottom_samples = bottom_boundary_samples(config)
initial_samples = initial_condition_samples(config)

Visually, this is how our domain looks like:

The neural network#

In this exercise, we employ a relatively simple feedforward neural network with one hidden layer, a few neurons, and $\tanh$ activation functions. While more complex architectures may be required for higher-dimensional problems, empirical evidence suggests that a simple network is sufficient for this particular case.

class PINN(nn.Module):
    def __init__(self):
        super(PINN, self).__init__()
        size = 15
        self.hidden_layers = nn.ModuleList([
            nn.Linear(2, size),
            nn.Linear(size, size),
        ])
        self.output_layer = nn.Linear(size, 1)

    def forward(self, inputs):
        x = inputs
        for layer in self.hidden_layers:
            x = torch.tanh(layer(x))
        x = self.output_layer(x)
        return x

It is also important to initialize the network weights properly, as this can affect the convergence of the solution. Unlike typical machine learning tasks that require large amounts of data, PINNs can often solve PDEs with relatively little (even synthetically generated) data. Therefore, global optimizers such as L-BFGS, which use the entire dataset when updating the weights, are particularly effective.

def initialize_weights(model):
    for layer in model.hidden_layers:
        if isinstance(layer, nn.Linear):
            nn.init.xavier_uniform_(layer.weight)
            nn.init.zeros_(layer.bias)

Training loop#

Now with all set, we can proceed to train the neural network.

def train_model(model, inputs, outputs, tags, sigma, r, iters=1000):
    optimizer = optim.LBFGS(
        model.parameters(),
        max_iter=iters,
        lr=1,
        line_search_fn='strong_wolfe',
        tolerance_grad=1e-10,
        tolerance_change=1e-10
    )

    loss_history = {'total_loss': []}

    def closure():
        optimizer.zero_grad()
        total_loss = 0

        pde_mask = (tags == 0).squeeze()
        boundary_mask = (tags == 1).squeeze()

        if pde_mask.any():
            pde_loss = pde_loss_f(model, inputs[pde_mask], sigma, r)
            total_loss += pde_loss

        if boundary_mask.any():
            boundary_loss = boundary_loss_f(
                model, inputs[boundary_mask], outputs[boundary_mask])
            total_loss += boundary_loss

        print(
            f'PDE Loss: {pde_loss.item()} | Boundary Loss: {boundary_loss.item()}', end='\r')
        total_loss.backward()
        loss_history['total_loss'].append(total_loss.item())
        return total_loss
    optimizer.step(closure)
    return loss_history

model = PINN().to(device)
initialize_weights(model)
loss_history = train_model(model, inputs, outputs,
                           tags, config['sigma'], config['r'], iters=10000)

PDE Loss: 0.0009885333711281419 | Boundary Loss: 838.0771484375
PDE Loss: 0.0006638792692683637 | Boundary Loss: 823.3145141601562
PDE Loss: 0.0025756016839295626 | Boundary Loss: 789.3040161132812
PDE Loss: 0.050156641751527786 | Boundary Loss: 644.5918579101562
PDE Loss: 1.827426552772522 | Boundary Loss: 227.68394470214844
PDE Loss: 60.65645217895508 | Boundary Loss: 698.048095703125
PDE Loss: 2.652808904647827 | Boundary Loss: 232.97764587402344
PDE Loss: 2.083289623260498 | Boundary Loss: 210.82704162597656
PDE Loss: 1.7472081184387207 | Boundary Loss: 206.156005859375
PDE Loss: 1.5691288709640503 | Boundary Loss: 204.77798461914062
PDE Loss: 1.6526966094970703 | Boundary Loss: 208.06614685058594
PDE Loss: 1.5836416482925415 | Boundary Loss: 203.8051300048828
PDE Loss: 1.5912162065505981 | Boundary Loss: 203.5250244140625
PDE Loss: 1.5972844362258911 | Boundary Loss: 203.3137969970703
PDE Loss: 1.6022659540176392 | Boundary Loss: 203.1529541015625
PDE Loss: 1.6077497005462646 | Boundary Loss: 202.9981689453125
PDE Loss: 1.6106784343719482 | Boundary Loss: 202.93020629882812
PDE Loss: 1.6158597469329834 | Boundary Loss: 202.8477020263672
PDE Loss: 1.6168268918991089 | Boundary Loss: 202.8390655517578
PDE Loss: 61.89354705810547 | Boundary Loss: 1451.9659423828125
PDE Loss: 5.331545352935791 | Boundary Loss: 331.5941467285156
PDE Loss: 2.0811665058135986 | Boundary Loss: 203.8124237060547
PDE Loss: 1.6579827070236206 | Boundary Loss: 205.06898498535156
PDE Loss: 1.6198105812072754 | Boundary Loss: 202.823974609375
PDE Loss: 1.6006227731704712 | Boundary Loss: 202.55661010742188
PDE Loss: 1.7255953550338745 | Boundary Loss: 198.8125762939453
PDE Loss: 5.182849407196045 | Boundary Loss: 250.0623321533203
PDE Loss: 1.8323962688446045 | Boundary Loss: 197.51071166992188
PDE Loss: 2.0654313564300537 | Boundary Loss: 195.2335205078125
PDE Loss: 2.2394537925720215 | Boundary Loss: 193.73471069335938
PDE Loss: 2.5798277854919434 | Boundary Loss: 191.0352020263672
PDE Loss: 2.8255228996276855 | Boundary Loss: 189.2777862548828
PDE Loss: 3.307149887084961 | Boundary Loss: 186.90187072753906
PDE Loss: 0.7587254643440247 | Boundary Loss: 606.8187866210938
PDE Loss: 23.960494995117188 | Boundary Loss: 868.080078125
PDE Loss: 3.9483423233032227 | Boundary Loss: 212.86212158203125
PDE Loss: 3.4382686614990234 | Boundary Loss: 186.672119140625

PDE Loss: 4.550113201141357 | Boundary Loss: 184.9908905029297
PDE Loss: 5.036774635314941 | Boundary Loss: 180.83509826660156
PDE Loss: 4.565935134887695 | Boundary Loss: 177.5693359375
PDE Loss: 3.8462483882904053 | Boundary Loss: 175.0547637939453
PDE Loss: 3.1329572200775146 | Boundary Loss: 173.59994506835938
PDE Loss: 2.72257661819458 | Boundary Loss: 172.73977661132812
PDE Loss: 2.659761905670166 | Boundary Loss: 171.90011596679688
PDE Loss: 2.696134567260742 | Boundary Loss: 170.10679626464844
PDE Loss: 3.212402820587158 | Boundary Loss: 164.4862518310547
PDE Loss: 4.7438883781433105 | Boundary Loss: 155.1752471923828
PDE Loss: 1.3468575477600098 | Boundary Loss: 306.3238220214844
PDE Loss: 5.098265171051025 | Boundary Loss: 148.5914306640625
PDE Loss: 5.189510822296143 | Boundary Loss: 142.33795166015625
PDE Loss: 6.759544372558594 | Boundary Loss: 130.19024658203125
PDE Loss: 5.763506889343262 | Boundary Loss: 472.1963195800781
PDE Loss: 6.961487293243408 | Boundary Loss: 128.68357849121094
PDE Loss: 7.583919525146484 | Boundary Loss: 124.89171600341797
PDE Loss: 44.83903503417969 | Boundary Loss: 12418.349609375
PDE Loss: 8.324292182922363 | Boundary Loss: 120.78887176513672
PDE Loss: 46.027496337890625 | Boundary Loss: 769.2361450195312
PDE Loss: 6.400981903076172 | Boundary Loss: 199.59625244140625
PDE Loss: 8.675870895385742 | Boundary Loss: 119.3843002319336
PDE Loss: 9.193161964416504 | Boundary Loss: 117.94184112548828
PDE Loss: 8.561506271362305 | Boundary Loss: 342.7839050292969
PDE Loss: 9.706830978393555 | Boundary Loss: 116.76162719726562
PDE Loss: 17.635847091674805 | Boundary Loss: 100.78704071044922
PDE Loss: 18.965042114257812 | Boundary Loss: 86.18229675292969
PDE Loss: 85.3647232055664 | Boundary Loss: 754.5050659179688
PDE Loss: 18.75090980529785 | Boundary Loss: 82.2439956665039
PDE Loss: 14.533609390258789 | Boundary Loss: 80.31694030761719
PDE Loss: 16.10525894165039 | Boundary Loss: 77.4521484375
PDE Loss: 8.242959022521973 | Boundary Loss: 84.63884735107422
PDE Loss: 18.89201545715332 | Boundary Loss: 68.65251159667969
PDE Loss: 27.24378204345703 | Boundary Loss: 64.9182357788086
PDE Loss: 21.79601287841797 | Boundary Loss: 63.65586853027344
PDE Loss: 16.340654373168945 | Boundary Loss: 66.29924774169922
PDE Loss: 16.95505714416504 | Boundary Loss: 63.1745719909668
PDE Loss: 10.863780975341797 | Boundary Loss: 77.29100799560547

PDE Loss: 15.00011920928955 | Boundary Loss: 63.09186935424805
PDE Loss: 14.797526359558105 | Boundary Loss: 60.94261932373047
PDE Loss: 13.916735649108887 | Boundary Loss: 57.907859802246094
PDE Loss: 14.736340522766113 | Boundary Loss: 53.086978912353516
PDE Loss: 14.248770713806152 | Boundary Loss: 46.853397369384766
PDE Loss: 12.069930076599121 | Boundary Loss: 39.899410247802734
PDE Loss: 9.78376579284668 | Boundary Loss: 34.79121017456055
PDE Loss: 8.604270935058594 | Boundary Loss: 29.606775283813477
PDE Loss: 10.718920707702637 | Boundary Loss: 19.78169822692871
PDE Loss: 3.694037437438965 | Boundary Loss: 95.33468627929688
PDE Loss: 5.941652774810791 | Boundary Loss: 20.554414749145508
PDE Loss: 5.1638312339782715 | Boundary Loss: 17.858423233032227
PDE Loss: 8.451488494873047 | Boundary Loss: 14.941847801208496
PDE Loss: 6.359133243560791 | Boundary Loss: 15.092984199523926
PDE Loss: 3.8142738342285156 | Boundary Loss: 15.899919509887695
PDE Loss: 4.454297065734863 | Boundary Loss: 15.358362197875977
PDE Loss: 4.100632190704346 | Boundary Loss: 15.216565132141113
PDE Loss: 3.4675536155700684 | Boundary Loss: 15.458770751953125
PDE Loss: 1.9045331478118896 | Boundary Loss: 16.59748649597168
PDE Loss: 2.2580933570861816 | Boundary Loss: 16.088939666748047
PDE Loss: 1.8886910676956177 | Boundary Loss: 15.800841331481934
PDE Loss: 1.7465623617172241 | Boundary Loss: 12.76977825164795
PDE Loss: 6.76230001449585 | Boundary Loss: 89.63126373291016
PDE Loss: 1.7672621011734009 | Boundary Loss: 12.351202964782715
PDE Loss: 1.9380217790603638 | Boundary Loss: 12.38087272644043
PDE Loss: 1.8398381471633911 | Boundary Loss: 11.711995124816895
PDE Loss: 4.72296667098999 | Boundary Loss: 84.70668029785156
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PDE Loss: 0.0009500279556959867 | Boundary Loss: 0.0012671280419453979
PDE Loss: 0.0009500287706032395 | Boundary Loss: 0.001267124549485743

Results#

Let’s see how the neural network performs in pricing the option. We will compare the network’s predictions against the true option price obtained from the Black-Scholes formula, but first let’s look at the loss evolution during training.

Benchmark#

How does it compare with the analytical solution? Below, we present a comparison between the analytical solution and the solution obtained via the neural network. As we know, the analytical solution for a European call option is given by

\[ C(S, t) = S\, \Phi(d_1) - K e^{-r(T-t)}\, \Phi(d_2), \]

where

\[ d_1 = \frac{\ln(S/K) + \left(r + \frac{\sigma^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}, \]

\[ d_2 = d_1 - \sigma \sqrt{T-t}, \]

and $\Phi$ denotes the cumulative distribution function (CDF) of the standard normal distribution. Below, we show the comparison between the analytical solution and the solution obtained with the neural network.

/var/folders/cp/l432p0ns1t38qmnyr_3l3r000000gn/T/ipykernel_16981/1113092756.py:2: RuntimeWarning:

divide by zero encountered in log

/var/folders/cp/l432p0ns1t38qmnyr_3l3r000000gn/T/ipykernel_16981/1113092756.py:2: RuntimeWarning:

divide by zero encountered in divide

Error (2-norm): 0.21%

We obtain an error of 0.21% against the exact solution without much tweaking! This demonstrates that even for a relatively simple problem such as pricing a European call option, the PINN approach can achieve a decent accuracy. This result is showcases the potential of this technique for more complex financial models, where analytical solutions are not available.